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1 into spans (we called span the intersection segment between a projected patch and a proxel array line). These spans are generated simultaneously for all the patches on the full processor array, as the computations are approximatively the same for each patch. After all the spans have been computed, they are sent to the processors where they appeared, in order to apply the depth buffer operations. Once all the spans have been sent, the depth buffer operations are performed, using the xnet communication network : the span is successively sent to each processor where it appears in order to apply the depth buffer operations. After scan conversion and depth buffer generation, the form factors are computed, using a proxel parallelism approach. The main advantage of this new approach is to increase dramatically the processors array efficiency, as all the processors apply the same operations on different patches, spans or proxels, taking fully advantage of the data parallelism inherent to SIMD computers. 7 CONCLUSION We described in this paper the implementation of two massively parallel approaches of the progressive radiosity algorithm, using projection methods as form factor approximation approach. The first one is based on proxel parallelism scheme for the scan conversion step, but the efficiency stays low, due to the small size of the projected patches compared to the size of the processor array. We derive a new approach that uses both object and proxel parallelism in the algorithm steps to get the optimal efficiency. This hybrid approach is currently under implementation, but it appears already that the processors efficiency will be much better than the previous one. However, care must be taken in order to limit the communications, that could increase the total computation time. REFERENCES [Cohen 85] [Cohen 88] [Goral 84] [Renaud 93a] [Renaud 93b] [Sillion 89] M.F. Cohen, D.P. Greenberg The hemicube : a radiosity solution for complex environments SIGGRAPH 85 Vol 19 No 3 pp 31-40, July 1985 M.F. Cohen, S.E. Chen, J.R. Wallace, D.P. Greenberg A progressive refinement approach to fast radiosity image generation SIGGRAPH 88 Vol 22 No 4 pp 75-84, August 1988 C.M. Goral, K.E. Torrance, D.P. Greenberg, B. Battaile Modeling the interaction of light between diffuse surfaces SIGGRAPH 84 Vol 18 No3 pp , July 1984 C. Renaud, F. Bricout, E. Leprêtre Hemispherical projection for progressive radiosity calculation on massively parallel architectures, 8th Eurographics Workshop on Graphics Hardware, pp 81-92, Barcelona, September 1993 C. Renaud Approches parallèles pour la radiosité PhD Thesis, University of Lille, France, October 1993 F. Sillion, C. Puech A general two-pass method integrating specular and diffuse reflection Computer Graphics Vol 23 No 3, pp , August 1989

2 proxels), and results shown (see figure 6.a) that moving the proxels to the projected patch, by using the global router, provides the lower communication time. Figure 6.b presents the scene we used for each of the measures that are presented in this paper, and contains about patches. FF update time (s) moving proxels (G.R.) moving patches (xnet) moving proxels (xnet) Resolution a) b) Figure 6 : average communication time for FF update (a) and test scene (b) Figure 7.a shows the average computation time for an emission step, for two different projection surfaces. Comparisons with sequential implementation underlines the small speedup obtained (2 to 4), due to the low efficiency of this approach. time in seconds Hemicube Disk resolution a) b) Figure7 : average computation times per emission step (a) average efficiency for a 128x128 processor array (b) Even by using bounding rectangles, the average number of proxels inside a patch projection stays small (see figure 7.b). 6 TOWARD AN HYBRID APPROACH resolution # of useful processors (/16384) In order to increase the efficiency of the scan conversion step, we study now a new approach, that exploits both object and proxel parallelism. In a first step, each patch is scan converted by the processor handling them. Geometrical transformations are applied simultaneously for all the patches, then they are transformed

3 version. After those two main stages, the form factor value can be updated taking into account proxel contributions. As the geometric transformations are the same for every patches, every array s processors treat a different patch. Each processor computes the patch projection parameters for the database it is in charge of. To execute the scan conversion each processor diffuses its own database to the full array, one patch after another. When receiving a patch, processors apply a depth-buffer mechanism for the proxels they manage that are covered by the projection. However, as a patch covers only a small part of the entire proxels set, efficiency stays low, a great part of the proxels should not have to be considered. The processors, also, only compute the proxels that are inside the bounding rectangle of the projected patch. Furthermore, as the number of processors is smaller than the number of proxels, these ones are distributed over the processors. We chose a cyclic distribution rather a block distribution, this former increasing the efficiency of the processor network. To update a patch form factor, all the proxels where the patch appears must be gathered. This generates many communications, as proxels and patches are distributed over the network. We can choose between moving the proxels to the corresponding patch, or moving the patch to the proxels it covers. We have compared the different communication schemes on the two communication networks available on the MP-1. By using the xnet, two approaches are available, either by moving the patches or the proxels. In the first solution, each patch has to go through all processors and to search in the local proxels if it is visible. In the second one, proxels have to be moved from the processor where they have been computed to the processor that handles the patch visible in the proxel. This is done by applying successive horizontal moves until to reach the column of the processor, and then vertical moves to get the processor line. These two communication ways are schematized on figure 5. xnet : patch to proxels xnet : proxels to patch global router : proxels to patch Figure 5 : communication schemes for form factor reduction By using the global router network, there is only one way to transmit the proxels to the visible patch (see figure 5). We have compared the execution time of each of these approaches for different resolutions (resolution N means that each face of the projection surface is discretized into NxN

4 As these methods are close to depth-buffer s, their parallelisation follows the same two schemes : object parallelism and pixel parallelism. 3 THE DEC MP-1 OVERVIEW We have developped our approaches on a DEC MP-1 computer. It is a massively parallel SIMD computer, fitted with an array of processor elements. The processor elements (64 Kb RAM each) are connected together via two communication networks : xnet for neighbour communications, and a global router for distant communications. 4 OBJECT PARALLELISM APPROACH In this approach, each processor manages a different patch, and determines the proxels covered by the patch it handles. A mechanism, called decider, has to be used in order to perform the depth buffer operations. Two main decider implantations are known in graphics hardware world : tree structure or pipe structure (see figure 3). decider :Object Processor Proxels C. C. C. Proxels C. : Comparison operator Figure 3 : Tree or pipe structured decider for the object approach The difficulty of this parallelisation scheme on a massively parallel computer resides in the decider implantation, as the depth buffer informations are distributed over all the processors. The evaluation of decider software implantation cost, even using reduction operations which seems to be the best solutions, allows us to conclude to the inefficiency of such an approach on the MP-1 [Renaud 93b]. 5 PROXEL PARALLELISM APPROACH Rather than allocating patches to processors, proxel parallelism approaches allocate a proxel (or a group of proxels) to each processor. Then, each patch is diffused to the processors, each one computing if the patch projection covers the proxels it handles (figure 4.a and 4.b). a) b) Patch Figure 4 : Patch diffusion (a) and inside projected patch proxels (b) The form factor evaluation requires a geometric transformation followed by a scan con-

5 However, both radiosity and progressive refinement radiosity are very time consuming. In this paper, we are interested in studying the ways to reduce the computation time of the progressive refinement radiosity by implementing it on a MP-1. In the next part, we present several projection algorithms used for form factor approximation. Part 3 summarizes the MP-1 characteristics. We describe in part 4 and 5 two common approaches for massively parallel scan conversion, and introduce in part 6 a new hybrid approach that provides efficient use of the processor array. 2 PROJECTIVE APPROACHES The main difficulty in form factor (FF) computation is to take into account the occlusions between a source and the patches of the scene. The first operation to perform is then to apply a hidden part elimination algorithm, in order to compute the exact part of the source energy reaching each patch. This process is similar to those which display images by using depth-buffer algorithms. However, several projection surfaces have been proposed, allowing to compute accurate energy exchanges and reduce the number of samples required. Some of these are the hemicube [Cohen 85], the single plane [Sillion 89], or the disk [Renaud 93]. Those surfaces are schematized on figure 2. z z z y y x x x the hemicube the single plane the disque Figure 2 : Projection surfaces for form factor computation Those surfaces are discretized into proxels (projected elements), at which are associated elementary form factors. The chosen surface is applied successively onto each emitted patch, in order to diffuse its energy into the scene. The FF evaluation is then cut into three distinct stages, described below : - the geometrical transformations, that prepare the patch for the next stage; - the projection step, that computes the proxels where a patch is visible, and apply depth-buffer operations in these proxels; - the form factor reduction, that adds all the elementary FF of the proxels where a patch is visible. Once all the patches have been projected and the form factors deduced, the source energy is distributed to the patches, and a new source patch is selected. The same process (projection - FF computation) is then applied for the new emitting patch, until all significant emitting patches have been treated. y

6 Two Parallel Schemes for Radiosity on the MP1 C. RENAUD, F. BRICOUT, E. LEPRETRE Laboratoire d Informatique Fondamentale de Lille U.R.A. 369 CNRS - USTL Villeneuve d Ascq Cedex - France {renaud, bricout, lepretre}@lifl.lifl.fr ABSTRACT Radiosity is a method which provides high quality images by taking into account diffuse to diffuse reflexions. Unfortunately, this method is very time-consuming, mainly due to the evaluation of geometric quantities, so called form factors. A great improvement can be obtained by implementing a specific designed algorithm on a large mesh-connected machine. The aim of this paper resides in a comparative study of the two possible parallelisation schemes: pixel approach and object approach. We also give some details about specific communication problems encountered in implementing the algorithms on the DEC MP1. Key-Words : Image Synthesis, Radiosity, Massively Parallelism, Projection Approaches 1 INTRODUCTION Radiosity [Goral 84] is a view independant global illumination method based on energy transfer evaluation between pairs of patches (assuming a planar patches modelisation). The patch s radiosities are obtained by solving a system of form-factor equations. Form factor is a purely geometric value representing the energy leaving a patch, reaching another. Form factor expression is given in figure 1.a, using the annotations of figure 1.b. 1 cosθ F ij ---- i cosθ j a) = A i da π r 2 j da i b) A i A j N i θ i r θ j Nj da j A j da i Figure 1 : form factor expression and geometry A i Form factors can be computed by using algorithms derived from depth-buffer ones, so called projective method [Cohen 85]. Progressive refinement method [Cohen 88] allows patch radiosities to be evaluated on the fly, rather than after system completion. This dramatically reduces the memory cost, and provides intermediate displays introducing some interactivity in the method.

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